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Document Type |
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Article In Journal |
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Document Title |
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Rank one operators and norm of elementary operators Rank one operators and norm of elementary operators |
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Document Language |
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Arabic |
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Abstract |
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Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples A = (A(1),..., A(n)) and B = (B-1,...,B-n) of elements in A, we define the elementary operator R-A,R-B on A by the relation R-A,R-B(X) = Sigma(n)(i=1) A(i) X B-i for all X in A. For a single operator A is an element of A, we define the two particular elementary operators L-A and R-A on A by L-A(X) = AX and R-A(X) = XA, for every X in A. We denote by d(R-A,R-B) the supremum of the norm of R-A,R-B(X) over all unit rank one operators on E. In this note, we shall characterize: (i) the supremun d(R-A,R-B), (ii) the relation d(R-A,R-B) = Sigma(i=1) parallel to A(i)parallel to parallel to B-i parallel to, (iii) the relation d(L-A - R-) = parallel to A parallel to + parallel to B parallel to, (iv) the relation d(LARB + LBRA) = 2 parallel to A parallel to parallel to B parallel to. Moreover, we shall show the lower estimate d(L-A - R-B) >= max{sup(lambda is an element of V(B)) parallel to A - lambda I parallel to, sup(lambda is an element of(A)) parallel to B - lambda I parallel to} (where V(X) is the algebraic numerical range of X in A). (C) 2006 Elsevier Inc. All rights reserved. |
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Journal Name |
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LINEAR ALGEBRA AND ITS APPLICATIONS |
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Volume |
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424 |
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Issue Number |
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1 |
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Publishing Year |
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2007 AH
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Added Date |
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Tuesday, June 17, 2008 |
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